Group #34199900 — subgroups

Subgroups — #34199900

Order 20224.

226 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098339	C₂	1	1	yes	maximal, characteristic
2	#23762163	C₂	1	1	yes	maximal, characteristic
3	#23761465	C₂	1	4	yes	maximal, complemented
4	C₇₉×#1566	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₉⋊C₁₆	C₄	1	2	yes	—
6	#19098620	—	2	1	no	—
7	#19098339	C₄	1	1	yes	characteristic
8	#19098339	C₄	1	1	yes	characteristic
9	#19098339	C₂²	1	2	yes	—
10	#19098316	—	2	1	no	—
11	#19098314	C₂²	1	2	yes	—
12	#423×C₇₉	—	2	1	no	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	#334×C₇₉	—	2	1	no	—
15	#334×C₇₉	C₂²	1	2	yes	—
16	C₂×C₇₉⋊C₁₆	—	2	4	no	—
17	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
18	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
19	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
20	C₂²⋊(C₇₉⋊C₈)	—	4	1	no	—
21	#14579072	—	4	1	no	—
22	C₂×M₁₆×C₇₉	—	4	1	no	—
23	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
26	C₄⋊C₈×C₇₉	—	2	2	no	—
27	C₄⋊C₈×C₇₉	—	4	1	no	—
28	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
32	C₂×C₇₉⋊C₈	—	4	2	no	—
33	C₇₉⋊C₁₆	—	2	2	no	—
34	C₇₉⋊C₁₆	M₁₆	1	4	yes	complemented
35	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
38	M₁₆×C₇₉	—	8	2	no	—
39	C₂×C₈×C₇₉	—	2	2	no	—
40	C₂×C₈×C₇₉	—	2	2	no	—
41	C₂×C₈×C₇₉	—	4	2	no	—
42	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
48	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
49	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂³×C₇₉	#106	1	1	yes	characteristic
51	C₂×C₄×C₇₉	—	2	2	no	—
52	C₂×C₄×C₇₉	—	2	1	no	—
53	C₂×C₄×C₇₉	—	2	1	no	—
54	C₂×C₄×C₇₉	—	4	1	no	—
55	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
56	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
58	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
61	C₈×C₇₉	—	2	1	no	—
62	C₈×C₇₉	—	2	1	no	—
63	C₈×C₇₉	—	8	2	no	—
64	C₈×C₇₉	#108	1	2	yes	—
65	C₈×C₇₉	C₂×M₁₆	1	2	yes	—
66	C₇₉⋊C₈	—	8	2	no	—
67	C₂²×C₇₉	—	2	1	no	—
68	C₂²×C₇₉	—	2	1	no	—
69	C₂²×C₇₉	#365	1	2	yes	—
70	C₂²×C₇₉	C₂×#106	1	1	yes	characteristic
71	C₄×C₇₉	—	2	1	no	—
72	C₄×C₇₉	—	2	1	no	—
73	C₄×C₇₉	—	2	1	no	—
74	C₄×C₇₉	—	2	1	no	—
75	C₄×C₇₉	—	2	1	no	—
76	C₄×C₇₉	#423	1	1	yes	characteristic
77	C₄×C₇₉	C₂×#108	1	1	yes	characteristic
78	#12019	—	79	1	no	maximal, Sylow
79	C₂×C₇₉	—	2	2	no	—
80	C₂×C₇₉	#1566	1	1	yes	characteristic
81	C₂×C₇₉	C₂×#365	1	1	yes	characteristic
82	C₂×C₇₉	#2155	1	1	yes	characteristic
83	C₂×#346	—	79	1	no	—
84	#1568	—	79	1	no	—
85	#1566	D₇₉	1	1	yes	characteristic
86	#1368	—	79	4	no	—
87	C₇₉	#12019	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	—	79	2	no	—
89	#432	—	158	1	no	—
90	#423	—	2	1	no	—
91	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
92	#346	—	79	2	no	—
93	#346	—	79	1	no	—
94	#346	—	79	1	no	—
95	#335	—	158	1	no	—
96	#334	—	2	1	no	—
97	#334	—	79	2	no	—
98	#334	C₂×D₇₉	1	2	yes	—
99	C₂×M₁₆	—	4	1	no	—
100	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
101	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
102	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
103	C₂×C₁₆	—	79	2	no	—
104	C₂×C₁₆	—	79	2	no	—
105	C₂×C₁₆	—	158	4	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	4	1	no	—
108	C₄⋊C₈	—	158	2	no	—
109	C₂²⋊C₈	—	316	1	no	—
110	#97	—	316	1	no	—
111	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
112	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
113	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
114	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
115	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
116	C₂²×C₄	#11929611	1	1	yes	characteristic
117	C₂²×C₄	#11929612	1	1	yes	characteristic
118	M₁₆	—	8	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	316	2	no	—
123	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
124	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
125	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
126	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
127	C₄²	Dic₇₉⋊C₄	1	1	yes	characteristic
128	C₄²	#11929611	1	1	yes	characteristic
129	C₄²	#11929612	1	1	yes	characteristic
130	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
131	C₁₆	—	79	4	no	—
132	C₁₆	—	158	2	no	—
133	C₂³	#14579092	1	1	yes	characteristic
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	#14579092	1	1	yes	characteristic
139	C₂×C₄	#14579092	1	1	yes	characteristic
140	C₂×C₄	#14579092	1	1	yes	characteristic
141	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
142	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
143	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	8	2	no	—
147	C₈	—	632	2	no	—
148	C₈	#14579096	1	2	yes	—
149	C₈	M₁₆×D₇₉	1	2	yes	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#19098385	1	2	yes	—
153	C₂²	C₂×#14579092	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	#19098586	1	1	yes	characteristic
160	C₄	C₂×#14579096	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	#23762155	1	1	yes	characteristic
163	C₂	C₂×#19098385	1	1	yes	characteristic
164	C₂	#23765174	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098339

C₂

yes

maximal, characteristic

#23762163

C₂

yes

maximal, characteristic

#23761465

C₂

yes

maximal, complemented

C₇₉×#1566

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₁₆

C₄

yes

—

#19098620

—

#19098339

C₄

yes

characteristic

#19098339

C₄

yes

characteristic

#19098339

C₂²

yes

—

#19098316

—

#19098314

C₂²

yes

—

#423×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#334×C₇₉

—

#334×C₇₉

C₂²

yes

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

—

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

Q₈

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

Q₈

yes

characteristic

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₇₉⋊C₈

—

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

M₁₆

yes

complemented

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

SD₁₆

yes

characteristic

C₂²×C₄×C₇₉

SD₁₆

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

C₄⋊C₄

yes

characteristic

C₄²×C₇₉

SD₁₆

yes

characteristic

C₄²×C₇₉

SD₁₆

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₂³×C₇₉

#106

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#108

yes

—

C₈×C₇₉

C₂×M₁₆

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#365

yes

—

C₂²×C₇₉

C₂×#106

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

—

D₇₉

yes

characteristic

#1368

—

C₇₉

#12019

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

#432

—

158

—

#423

—

C₂×C₄×C₈

C₂×D₇₉

yes

—

—

—

—

158

—

#334

—

#334

—

#334

C₂×D₇₉

yes

—

C₂×M₁₆

—

100

C₂²×C₈

Dic₁₅₈

yes

characteristic

101

C₂²×C₈

C₄×D₇₉

yes

characteristic

102

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

158

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

158

—

109

C₂²⋊C₈

—

316

—

110

#97

—

316

—

111

C₄×C₈

Dic₁₅₈

yes

characteristic

112

C₄×C₈

C₄×D₇₉

yes

characteristic

113

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

114

C₄×C₈

C₂²×D₇₉

yes

characteristic

115

C₂²×C₄

Dic₇₉⋊C₄

yes

characteristic

116

C₂²×C₄

#11929611

yes

characteristic

117

C₂²×C₄

#11929612

yes

characteristic

118

M₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

316

—

123

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

124

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

125

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

126

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

127

C₄²

Dic₇₉⋊C₄

yes

characteristic

128

C₄²

#11929611

yes

characteristic

129

C₄²

#11929612

yes

characteristic

130

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

131

C₁₆

—

132

C₁₆

—

158

—

133

C₂³

#14579092

yes

characteristic

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#14579092

yes

characteristic

139

C₂×C₄

#14579092

yes

characteristic

140

C₂×C₄

#14579092

yes

characteristic

141

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

142

C₂×C₄

C₂×#11929611

yes

characteristic

143

C₂×C₄

C₂×#11929612

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

632

—

148

C₈

#14579096

yes

—

149

C₈

M₁₆×D₇₉

yes

—

150

C₂²

—

151

C₂²

—

152

C₂²

#19098385

yes

—

153

C₂²

C₂×#14579092

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

#19098586

yes

characteristic

160

C₄

C₂×#14579096

yes

characteristic

161

C₂

—

162

C₂

#23762155

yes

characteristic

163

C₂

C₂×#19098385

yes

characteristic

164

C₂

#23765174

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14579092
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1566	C₂
socle	C₂²×C₇₉	C₂×#106
prime radical	C₂³×C₇₉	#106
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#1566	D₇₉
hypocenter	C₇₉	#12019
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1566

subgroup

G/H

center

C₂²

C₂×#14579092

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1566

C₂

C₂²×C₇₉

C₂³×C₇₉

C₄×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1566

N	H ≅ G/N	type
3. #23761465	C₂	semidirect
34. C₇₉⋊C₁₆	M₁₆	semidirect
87. C₇₉	#12019	semidirect

H ≅ G/N

type

3. #23761465

C₂

semidirect

34. C₇₉⋊C₁₆

M₁₆

semidirect

87. C₇₉

#12019

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#19098339	2
#23762163	2
#23761465	2
C₇₉×#1566	2
#12019	79

subgroup

index

C₇₉×#1566

Subgroups — #34199900

Conjugacy classes of subgroups

Complements

Maximal subgroups